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UNIVERSITY OF COLOKADO STUDIES 



of this kind. In the present chapter I shall discuss the loxodromics 

 of the torus and their properties of closure.' 



2. Designating by u and v the angles, which, in Fig. 12, de- 

 termine the position of a point B on the surface of a torus, the square 

 of a linear element on the surface has the form. 



d8^:=. (R+r sin vyi^du^-^-dv^')^ 

 where v/, ='\i and i\ = T" ^^^ 



R 



-\- sin V 



(1) 



(2) 



Fig 12. 



R and r are respectively the radii of the axial circle and of a 

 meridian of the torus. As it is well known, by means of the ex- 

 pressions (2), the points of the torus are conformally transformed 

 into the points of a rectangle. Considering exclusively the case 

 where R>r, the integral v^ has the value 



(1) A preliminary statement concerninff orthogrraphic loxodromics on atoms was made by 

 the author in a paper, "Ueber orthoeonale Systeme und einigre technische An- 

 wendungen," which appeared in the progrram of the Polytechnic of Biel, 1898. 



An article treating of the same subject appeared in the American Math. Monthly, Vol. VI. 

 p. 136-138. 



